We know the definition of subspace which is:

If $\displaystyle W$ is a set of one or more vectors from a vector space $\displaystyle V$, then $\displaystyle W$ is a subsace of $\displaystyle V$ if and only if the following conditions hold:

(a) If $\displaystyle u$ and $\displaystyle v$ are vectors in $\displaystyle W$, then $\displaystyle u + v$ is in $\displaystyle W$.

(b) If $\displaystyle k$ is any scalar and $\displaystyle u$ is any vector in $\displaystyle W$, then $\displaystyle ku$ is in $\displaystyle W$.

Also the definition of $\displaystyle span(S)$ is:

If $\displaystyle S = \{v_1, v_2, ..., v_r\}$ is a set of vectors in a vector space $\displaystyle V$, then the subspace $\displaystyle W$ of $\displaystyle V$ consisting of all linear combinations of the vectors in $\displaystyle S$ is called the space spanned by $\displaystyle v_1, v_2, ...,v_r$, and we say that the vectors $\displaystyle v_1, v_2, ..., v_r$ span $\displaystyle W$. To indicate that $\displaystyle W$ is the space spanned by the vectors in the set $\displaystyle S = \{v_1, v_2,..., v_r\}$, we write:

$\displaystyle W = span(S)$ or $\displaystyle W = span\{v_1, v_2,..., v_r\}$

Now the problem I'm having is how is it possible that $\displaystyle span(S)$ is a subspace? I can't make a connection with the definition and the proof.

Is it possible to explain the proof of why $\displaystyle span(S)$ is a subspace?